Pacific Journal of Mathematics

BAIRE SPACES AND HYPERSPACES

ROBERT ALLEN MCCOY

Vol. 58, No. 1 March 1975 PACIFIC JOURNAL OF MATHEMATICS Vol. 58, No. I, 1975

BAIRE SPACES AND HYPERSPACES

ROBERT A. MCCOY

This paper examines the question as to when the hyperspace of a Baire is a , and related questions. An answer is given in terms of a certain product space's being a Baire space.

A hyperspace of a space X is the space of closed subsets of X under a natural . In this paper we investigate what happens to Baire spaces in the formation of hyperspaces. The first section is devoted to a discussion of the basic concepts. In the second section some characterizations of Baire spaces are given which will be useful while working with hyperspaces, the study of which occurs in the third section. In particular, we shall be primarily concerned with two basic questions. If X is a Baire space, when is the hyperspace of X a Baire space? If the hyperspace of X is a Baire space, when is X a Baire space? We also look briefly at Baire spaces in the strong sense and pseudo-complete spaces.

1. Basic definitions and properties. A Baire space is a space in which every countable intersection of dense open subsets is dense. It can also be defined as a space such that every nonempty open subspace is of second category. The usual definition of a space of first category is one which can be written as a countable union of nowhere dense subsets (i.e., subsets whose closures have no points). A space is of second category then if it is not of first category. Also second category spaces can be characterized as spaces in which every countable intersection of dense open subsets is nonempty. We now Hst a few of the properties which the Baire space concept enjoys. Of course the gives a sufficient condition for a space to be a Baire space. That is, every complete is a Baire space. Also every locally compact is a Baire space. Clearly every nonempty open subspace of a Baire space is a Baire space. In fact a space is a Baire space if and only if every point has a neighborhood which is a Baire space. A useful spaces is that every space which contains a dense Baire subspace is a Baire space. The question as to which products of Baire spaces are Baire spaces has been a difficult one. There have been very few different examples

133 134 ROBERT A. McCOY discovered of Baire spaces whose product with itself is not a Baire space (see [9] and [6]). In §3, we shall be interested in having Xω be a Baire space. This then will be the case for most known Baire spaces X. In fact it will always be true that Xω is a Baire space if X is a Baire space having a countable base. On the otherhand if Xω is a Baire space, then X will always be a Baire space, since an open continuous image of a Baire space is a Baire space. A thorough investigation of the properties of Baire spaces can be found in [5]. When working with hyperspaces, we shall use the notation and terminology in [7]. Briefly, if X is a with topology 3Γ, 2X denotes the set of all nonempty closed subsets of X and ^(X) denotes the set of all nonempty compact subsets of X If Ui9 , Un are subsets of X, then

<£/„.. , and

A n Ui ^ 0 for each f = 1, , n\ .

The finite topology (or Vietoris topology) on 2X, denoted by 2*, is the topology generated by the base

The topology S'c on %{X) is the topology generated by the base c $ = {

C if and only if UΓ-i UtC U?>, Vi

and for each V} there exists a ty such that ULCVr, and

C\{(UU , Un)) = (Uu , ί/n> (where C1U and C7 both de- note the closure of U).

We shall occasionally be concerned with quasi-regular spaces, that is, spaces such that every nonempty contains a closed subset with nonempty interior. It is not difficult to see that if X is quasi- regular, then 2X and ^(X) are quasi-regular. BAIRE SPACES AND HYPERSPACES 135

2. Characterizations of Baire spaces. In §3, we shall need ways of looking at Baire spaces other than our definition. In this section, we establish the needed characterizations. Let X be an arbitrary topological space. By a pseudo-base for X is meant a collection of nonempty open subsets of X such that each nonempty open subset of X contains a member of this collection. Let 5δ be a pseudo-base for X, Define

S(X, 38) = {/: 3δ -> 9t \f(U) C U for every U G 3δ}, and

RS(X,®) = {f: 95-^28 \f(U)CU for every l/e&}.

If U E m and /, g e 5(X, B) or Jί5(X, 38), define

[L7,/,g], = g(l/), and for i > 1, , {/([[/,/, gL-i), if i is even, L^,7,^JΓfί f . = l'.'rr/^g]^ if/is odd.

The following theorem can be found in [5], and similar theorems can be found in [6] and [8]

THEOREM 2.1. The following are equivalent. (i) X is a Baire space. (ii) There exists a pseudo-base 5δ for X such that for every U G 5δ and f G S(X, $), f/im? ex/sίs α g G 5(X, Sδ) swc/i that

(iii) For βi ^ry pseudo-base Sδ /or X and ei ery t7 G $β and f G

δ), there exists a g G5(X,Sδ) such that ΠΓ=1 [ί/,/,g]( ^0. There is an analogous result for spaces of second category.

THEOREM 2.2. 77ιe following are equivalent. (i) X /s o/ second category. (ii) 77zere ex/sίs α pseudo-base Sδ /or X ,swc/ι that for every f G 5(X, Sδ), ίΛ^r^ ejc/5ί5 α [7 G Sδ and g G S(X, Sδ)

(iii) For every pseudo-base $β for X and every f G S(X, Sδ),

[/GSδ and gES(X^) such that Π7-ι[U9f,g]ιj£0.

The characterization which we actually use in the next section is given in the following theorem. 136 ROBERT A. McCOY

THEOREM 2.3. Let X be a quasi-regular space, and let 38 be a pseudo-base for X. Then the following are equivalent. (i) X is a Baire space. (ii) For every UEB and f E S(X,S3), there exists g G #S(X, S3) such that nr-i[l/,/,g]i^0. (iii) For every UE@ and /Gl?S(X, S3), there exists g E

RS(X93S) such that Π7=, [t/,/,g],^0. (iv) For every U E 3$ and f E RS(X, S3), there exists g E S(X,S3) such that n7-ΛU,f,glέ0.

Proof For every t/efl5, /,?eS(X,») with f(U)?U and g(U)CU, define /; g G £S(X, S3) as follows. Since X is quasi-regular, for every V6«, there exists /(V)eB such that ]{y)Cf(V). If

V = /([J7, JUk-,) for some Λ = 1,2, , let g(V) = [£/,/,g]2n+1; if V = 17, let g(\Q = g(V); and otherwise let g(V) be an element of $ such

thatΛ g(V)CV. Now observe that [t/,/,g], = g(U) = g(U) - [t/,/,g],. Suppose that [t/,/,gk-i = [l/,/,gk-i for every k =

l, ;,π. Then [ϋJ,ίL, = ί(ffflίJ,ί]taJ = ίpl/,/,gU)«

[[/,/,g]2π+1. Therefore by induction [L/,/,g]2n_, = [l/,/,^]2«-i for every n. To see that (i) imples (ii), suppose that (ii) does not hold. Then there exists U E 53 and / G S(X, S3) such that for every g G #S(X, S3), nΐ=ί[UJ,gl =0. Let g be an arbitrary element of S(X,$l We may assume without loss of generality that f(U)^U and g(U)C U. Now nr=,[l/,/,gL = nΓ=i[t/,/,gL =0, so that X cannot be a Baire space by Theorem 2.1. Clearly (ii) implies (iii) and (iii) implies (iv), so that it remains to show that (iv) implies (i). Suppose X is not a Baire space. Then there exists ί/ε« and /G5(X,S8) such that for every gG5(X, S3), Π Γ-i [U, /, g]i = 0. Let g be an arbitrary element of 5(X, S3). Again we may assume without loss of generality that f(U)^U and g(U)C

U. Then nr=,[U,/,g]i=nr=i[ί/,/J]ί=0, which establishes the negation of statement (iv) by Theorem 2.1 again. Just as Theorem 2.1 has Theorem 2.2 as an analog for spaces of second category, Theorem 2.3 has its analog for spaces of second category, which we shall not state since it can easily be induced.

3. Applications to hyperspaces. First we establish that

for a Γrspace X, if the hyperspace of X is a Baire space, then X must be a Baire space. Since the following two theorems have essentially the same proofs, we only give the proof for the second theorem. BAIRE SPACES AND HYPERSPACES 137

THEOREM 3.1. If^(X) is a Baire space (space of second category, respectively), then X is a Baire space (space of second category, respectively).

X THEOREM 3.2. IfXis a Trspace and 2 is a Baire space (space of second category, respectively), then X is a Baire space (space of second category, respectively).

Proof. Suppose that X is not a Baire space. Then there exists an open U in X and a {C/J of dense open subsets of X such that

U n (n Γ-i Ut) = 0. Let G = <[/>, and for each i, let G, = (l//>. To X see that each G, is dense in 2 , let (Vι, — 9Vn) be an element of mx. Now each VJΠί/,^0 for / = 1, ,n. So let x, EVjΓiUi for each such j, and let A ={JC,, ,jcn}. Since X is a ΓΓspace, A E X 2 . Also A is easily seen to be in G,? Π (Vx, , Vn), so that each G, is dense in 2X. Finally, it is clear that G Π (Π 7=i G, ) = 0, so that 2* could not be a Baire space.

COROLLARY 3.3. IfXis a first category space, then so is ^(X). If X in addition X is a Trspace, then 2 is a first category Trspace.

The Γj-space hypothesis in Theorem 3.2 cannot be omitted since the space of natural numbers, N, with the topology consisting of right rays is a space of first category, while 2N has the indiscrete topology, so is a Baire space. Note that this is also an example of a space X such that 2X is quasi-regular, but X is not quasi-regular. Before examining the converses of Theorem 3.1 and 3.2, we wish to introduce a new space. Let Xω denote the Cartesian product of a countably infinite number of copies of X. Instead of putting the usual product topology on Xω, we shall need to consider a topology if* on Xω which lies strictly between the product topology and the box topology. If Uu- ,Un are subsets of X, let U(U{, , Un) denote the

subset (ΠΓ-, ϋi) x (ΠΓ=Π+1 (U U Uj)) of X\ Let

® * = {Π(l/,, , Un) I 17, is open in X, i = 1, , n}.

Finally, let 5"* be the topology on Xω generated by 35*.

ω LEMMA 3.4. The family ffl* is a base for (X , J*).

Proof. Let U = U(UU- , Un) and V = Π(Vx, , Vm) be arbitrary members of ^*. We may suppose that n^m. Then 138 ROBERT A. McCOY

u n v = π (u, n v,, , um n vm, um+l n (y v,), , un n(,Q which is a member of S3*. Lemma 3.4 now gives us the following fact.

ω LEMMA 3.5. Each projection map on (X , 3~*) is continuous and open. It is easy to see that open continuous functions preserve Baίre spaces and spaces of second category (see [4]). The next lemma then follows from Lemma 3.5.

ω LEMMA 3.6. // (X , 2Γ*) is a Baire space (space of second cate- gory, respectively), then X is a Baire space (space of second category, respectively).

A modification of the proofs of 2.5 and 2.6 in [9] establishes the following fact.

LEMMA 3.7. // X is a Baire space (space of second category, respectively) having a countable pseudo-base, then (Xω,β*) is a Baire space (pace of second category, respectively).

We now introduce some notation which will be used in the proof of the next two theorems. We shall be working with the three spaces (XVT*), (2X,2^), and (<€(X),?ΓC) having bases $*,$*, and ®c, x respectivley. If U = 11(17,, •• , Un) G S3, then define U = c (Ux,- ,Un), which is an element of 33*, and define U = c (Ul,'",Un)n €(X), which is an element of ST. On the other hand if x c c U = (Uλ,' ',Un)E:® or U = (Uι,'-,Un)n €(X)E^ , define 17* =

Π({7,, , Un), which is an element of S3*. Also for each /G RS(Xω,®*), define fx ERS(2X, ®x) and f ERS(C€(X),^C) as x follows. If U = (Uι, -,Un) is an arbitrary element of ® , define x x f (U) = (f(U*)) , and if U = , where m^n and for each i^l, ,n,

V; C Uh Now if 17 = Π(C7,, , Un) is an arbitrary element of S3*, then define /*(17) = (f(Ux))*. A similar definition is to be given for /* if f<ΞRS(<β(X),®c). BAIRE SPACES AND HYPERSPACES 139

ω THEOREM 3.8. If X is quasi-regular and (X , ZΓ*) is a Baire space (space of second category, respectively), then 2X is a Baire space (space of second category, respectively).

Proof Let U e ®x and let / e RS(2X, ®x). Since (Xω, ^*) is a Baire space, by Theorem 2.3, there exists g GRS(Xω,ffl*) such that ω x nΓ-i[t/*,/*,g]i contains some element (JC() of X . Now [U,f,g ]2 = x x f(g (U)) = f((g(U*)) ), and [t/*,/*,g]2 = (f((g(U*))x))*. Also for each n > 1,

x x ,g ]fn-2)) ), and

[U*,f*,g]u = f*(g([U*9f*,g]u-ύ)

x So that by induction, [E/*,/*,g]2n = ([t/,/,g ]2π)* for every

n. Therefore since (JC,)G [C/*,/*,g]2n for every n, then {x{}e x x X [C/,/,g ]2n for every n. Thus ΠΓ=ι [U,f,g l^0, so that 2 must be a Baire space by Theorem 2.3 again.

COROLLARY 3.9. // X is a quasi-regular Baire space (space of second category, respectively) having a countable pseudo-base, then 2X is a Baire space (space of second category, respectively).

If we consider the space ^(X) instead of 2X, then we obtain the converse of Theorem 3.8.

THEOREM 3.10. If X is quasi-regular and ^(X) is a Baire space (space of second category, respectively), then (Xω, 3~*) is a Baire space (space of second category, respectively).

Proof. Let U E2&* and let f<ΞRS(Xω, 38*). Since %(X) is a Baire space, by Theorem 2.3, there exists a g ^RS(C€(X), ®c) such that ΠΓ=i[£/c,/c,g]/ contains some element A of ^(X). Now c c c c c [UJ,g*]2 = f(g*(U)) = f((g(U ))*), and [U ,f ,g]2 = f (g(U )) = /((g(l/c))*))c Also for each n > 1,

[U,f,g*]2n=f(g*([U,f,g*]2n-2)) c = f((g([U,f,g*] 2n-2))*), and c c c c c [U ,f ,g]2n=f (g([U J ,g]2n-2)) 140 ROBERT A. McCOY

c So that by induction, [ JJ\ f\ g ]2n = ([ [/, /, g *]2n ) for every n. Suppose e c that for each n, [U ,f ,g]2n = <£/?, , t/^jn «(*). Then each

[t/,/,g*]2n=([/ΐ, ,C/^)), so that m(l)Sm(2)^ , and for each l n / = l, ,m(n), Ur CU t. Let i be a fixed positive integer. If there exists an n such that i^m(n), let n(/) be the smallest such n. In this case, for each j = n(i)y n(i) + 1, , let a[^A Π U\. Otherwise, for each j = 1,2, , let a[ E A Π(UT^ U{). In either case, since A is compact,

{a\ \j = 1,2, •} has a cluster point, say ah Now letting i vary, this ω defines the point (at) E X . It can be seen that (α( ) E ΠΓ=i [{/,/, g*]f, so that (Xω, &*) must be a Baire space by Theorem 2.3 again. We might observe that when X is a quasi-regular space, Theorem 3.1 now follows from Theorem 3.10 and Lemma 3.6. Using Theorem 3.10, we can now obtain an example of a metric Baire space K such that ^(K) is not a Baire space. In [9], Oxtoby gave an example of a Baire space whose square is not a Baire space. This space unfortunately needs the continuum hypothesis in its construction. Later in [6], Krom constructed a new space K from Oxtoby's example which is a metric Baire space whose square is not a Baire space. Now if X2 is not a Baire space, it is easy to see by a modification of Lemma 3.6 that (Xω, 5"*) is not a Baire space. Thus by Theorem 3.10, ^(K) is not a Baire space. It would be of interest to know whether 2K is a Baire space, where K is Krom's example discussed above. One is tempted to try to answer this by investigating the relationship between 2X and ^(X) in terms of being a Baire space. First, since any extension of a Baire space is a Baire space, we immediately get the following.

THEOREM 3.11. // ^(X) is a Baire space, then so is 2*.

On the other hand, Aarts and Lutzer in [1] gave a test to determine when a dense subspace of a Baire space is a Baire space. A modifica- tion of this theorem (see [5]) is that if X is a dense subspace of the Baire space y, then X is a Baire space if and only if every somewhere dense

Gδ- subset of Y intersects X (a set being somewhere dense if its closure has nonempty interior). Therefore, as partial converse of Theorem 3.11, we have the following.

X THEOREM 3.12. // 2 is a Baire space and every somewhere dense Gδ-subset of 2X intersects ^(X), then ^(X) is a Baire space.

However, the full converse of Theorem 3.9 is false, as we shall now extablish. Let P be a dense Baire subspace of the irrationals with the BAIRE SPACES AND HYPERSPACES 141 usual metric topology having the property that every compact subset has an isolated point (see for example [1], example 2.4; or [5], Theorem 2.6). Let $ be the set of all intervals in P having rational end points and diameter less than one. For each pairwise disjoint collection

{/,,• -,Ik} of elements of & and for each natural number n, let

where for each / = 1, , fc, the JJ's are pairwise disjoint intervals such that I = U "=i I{ and diam /{ = IIn diam ί for each / = 1, , n. Now for each n, let Gn = U{Gn(Iu -,/*)!/,, ,Ik are pairwise disjoint elements of ^}, which is a dense open subset of 2P. To see that

(Π:=1Gn)Π^(P) = 0, let AG^iP). Now A must contain some isolated point x. Let m be a natural number such that no other point of

A is within 3/m of x. If for some Iu , Ifc ε *0, A G Gm(Iu * * *,/*), then x E /{ for some « = 1, , k and j = 1, , m. But then for some /' = 1, , m, A Π I\' = 0 — which contradicts A being in

Gm (/„ , Ik). Therefore (n:=ιGn)Π <€{P) = 0, so that

X THEOREM 3.13. Let X be a TΓspace. If either 2 or «(X) is a Baire space in the strong sense, then so is X.

A space X is called pseudo-complete if it is quasi-regular and there exists a sequence {{%J of pseudo-bases for X such that if for each

i = 2,--,UiEdii and Ui+ι C Ui9 then Π Γ=i U^0. It is not difficult to see that every pseudo-complete space is a Baire space. Properties of pseudo-complete spaces can be found in [9] and [1]. One such interesting property of pseudo-complete spaces (found in the latter reference) is that for a metric space X, X is pseudo-complete if and only if it has a dense completely metrizable subspace. 142 ROBERT A. McCOY

X THEOREM 3.14. If X is pseudo-complete, then so is 2 .

Proof. Let {53,} be a sequence of pseudo-bases for X such that if for each i = 1,2, , [/f G % and Ui+ι C Uh then Π Γ=i C/f φ 0. For each

ί, let 38f = {<[/,, , l/π')| [/,, ••-,[/„£ 98f}, which is a pseudo-base for

2*. Now for each i, let Gf =

{jcy 11 ^j < L}. NowΛ G d for every i, so that Λ G Gί+1 CG, for every i. Therefore A G ΠΓ=i G,, so that 2* is pseudo-complete. The converse of Theorem 3.14 is easily seen to be false since the space N of natural numbers with the right ray topology is not quasi- regular, while 2X has the indiscrete topology and is hence pseudo- complete.

REFERENCES

1. J. M. Aarts and D. J. Lutzer, Pseudo-completeness and the product of Baire spaces, Pacific J. Math., 48 (1973), 1-10. 2. , The product of totally nonmeagre spaces, Proc. Amer. Math. Soc, 38 (1973), 198-200. 3. N. Bourbaki, , part 2, Addison-Wesley Publishing Company, Reading, Massachusetts, 1966. 4. Z. Frolik, Remarks concerning the invariance of Baire spaces under mappings, Czech. Math. J., 11 (I960, 381-385. 5. R. C. Haworth and R. A. McCoy, Baire spaces, (to appear). 6. M. R. Krom, Cartesian products of metήc Baire spaces, Proc. Amer. Math. Soc, 42 (1973), 588-594. 7. E. Michael, on spaces of subsets, Trans. Amer. Math. Soc, 71 (1951), 152-182. 8. J. C. Oxtoby, The Banach-Mazur game and Banach Category Theorem, Contribution to the theory of games Vol. Ill, (Annals of Mathematics Studies number 39), Princeton University Press, Princeton, 1957.

9. , Cartesian products of Baire spaces, Fund. Math., 49 (1961), 157-166.

Received February 28, 1974 and in revised form May 28, 1974.

VPI & SU, BLACKSBURG, VIRGINIA CONTENTS

Zvi Artstein and John A. Burns, Integration of compact set-valued functions 297 J. A. Beachy and W. D. Blair, Rings whose faithful left ideals are cofaithful 1 Mark Benard, Characters and Schur indices of the unitary reflection group [321]3 309 H. L. Bentley and B. J. Taylor, Wallman rings 15 E. Berman, Matrix rings over polynomial identity rings II 37 Simeon M. Berman, A new characterization of characteristic functions of absolutely continuous distributions 323 Monte B. Boisen, Jr. and Philip B. Sheldon, Pre-Prufer rings 331 A. K. Boyle and K. R. Goodearl, Rings over which certain modules are injective 43 J. L. Brenner, R. M. Crabwell and J. Riddell, Covering theorems for finite nonabelian simple groups. V 55 H. H. Brungs, Three questions on duo rings 345 Iracema M. Bund, Birnbaum-Orlicz spaces of functions on groups ....351 John D. Elwin and Donald R. Short, Branched immersions between 2- ofhigher topological type 361 J. K. Finch, The single valued extension property on a 61 J. R. Fisher, A Goldie theorem for differentiably prime rings 71 Eric M. Friedlander, Extension functions for rank 2, torsion free abelian groups 371 J. Froemke and R. Quackenbusch, The spectrum of an equational class of groupoids 381 B. J. Gardner, Radicals of supplementary semilattice sums of associative rings 387 Shmuel Glasner, Relatively invariant measures 393 G. R. Gordh, Jr. and Sibe Mardesic, Characterizing local connected- ness in inverse limits 411 S. Graf, On the existence of strong liftings in second countable topological spaces 419 S. Gudder and D. Strawther, Orthogonally additive and orthogonally increasing functions on vector spaces 427 F. Hansen, On one-sided prime ideals 79 D. J. Hartfiel and C. J. Maxson, A characterization of the maximal monoids and maximal groups in βx 437 Robert E. Hartwig and S. Brent Morris, The universal flip matrix and the generalized faro-shuffle 445 aicJunlo ahmtc 95Vl 8 o 1 No. 58, Vol. 1975 Mathematics of Journal Pacific Pacific Journal of Mathematics Vol. 58, No. 1 March, 1975

John Allen Beachy and William David Blair, Rings whose faithful left ideals are cofaithful ...... 1 Herschel Lamar Bentley and Barbara June Taylor, Wallman rings ...... 15 Elizabeth Berman, Matrix rings over polynomial identity rings. II ...... 37 Ann K. Boyle and Kenneth R. Goodearl, Rings over which certain modules are injective ...... 43 J. L. Brenner, Robert Myrl Cranwell and James Riddell, Covering theorems for finite nonabelian simple groups. V ...... 55 James Kenneth Finch, The single valued extension property on a Banach space ...... 61 John Robert Fisher, A Goldie theorem for differentiably prime rings...... 71 Friedhelm Hansen, On one-sided prime ideals ...... 79 Jon Craig Helton, Product integrals and the solution of integral equations ...... 87 Barry E. Johnson and James Patrick Williams, The range of a normal derivation ...... 105 Kurt Kreith, A dynamical criterion for conjugate points...... 123 Robert Allen McCoy, Baire spaces and hyperspaces ...... 133 John McDonald, Isometries of the disk algebra ...... 143 H. Minc, Doubly stochastic matrices with minimal permanents ...... 155 Shahbaz Noorvash, Covering the vertices of a graph by vertex-disjoint paths...... 159 Theodore Windle Palmer, Jordan ∗-homomorphisms between reduced Banach ∗-algebras ...... 169 Donald Steven Passman, On the semisimplicity of group rings of some locally finite groups ...... 179 Mario Petrich, Varieties of orthodox bands of groups ...... 209 Robert Horace Redfield, The generalized interval topology on distributive lattices ...... 219 James Wilson Stepp, Algebraic maximal semilattices ...... 243 Patrick Noble Stewart, A sheaf theoretic representation of rings with Boolean orthogonalities ...... 249 Ting-On To and Kai Wing Yip, A generalized Jensen’s inequality ...... 255 Arnold Lewis Villone, Second order differential operators with self-adjoint extensions ...... 261 Martin E. Walter, On the structure of the Fourier-Stieltjes algebra ...... 267 John Wermer, Subharmonicity and hulls ...... 283 Edythe Parker Woodruff, A map of E3 onto E3 taking no disk onto a disk ...... 291